# 3D Geometry Proof by Contradiction /Contrapositive (high school)

Could someone evaluate my work?

A plane perpendicular to one of 2 parallel lines is perpendicular to the other line also.

My two column proof so far:

1. Let AB || CD and AB be perpendicular to plane p. Reason: Given
2. Assume by way of contradiction that CD is not perpendicular to p. Reason: Assumption
3. Let CX be drawn so that CX is perpendicular to p at point X. Reason: Assumption
4. AB and CD form plane q. Reason: Two lines determine a plane (analogy to two points determine a line)
5. CX and CD are both parallel to AB Reason: Steps 1,2 and 4
6. CX = CD. Reason: ?

I just know at the last step I have some kind of a contradiction which leads me to conclude that CD is perpendicular to p. Would I also have to use the Fifth postulate (parallel postulate) somewhere?

• If by parallel you mean they do not intersect then this is false. Take $l_1$ to be the $x$ axis and $l_2$ to be the $y$ axis shifted by one unit along the $z$ axis. Clearly $l_1$ and $l_2$ do not intersect. Now, the plane $xz$ is perpendicular to $l_2$ but this does not imply that it is perpendicular to $l_1$, it even contains $l_1$. – fidbc Mar 27 '13 at 20:51
• This is Theorem 10.10 in my book. – Person Mar 27 '13 at 20:52
• @fidbc Judging from a brief search, it seems like it is standard to apply "parallel" only to coplanar lines. – rschwieb Mar 27 '13 at 21:05
• @Person I don't think you would label #3 as "assumption", I think you would appeal to whatever result you have that says you can drop a perpendicular line from $C$ to the plane. Do you really have to use proof by contradiction? It seems fairly easy to do directly, instead. – rschwieb Mar 27 '13 at 21:12
• Are you allowed to use "distinct planes that intersect must intersect in a line"? How do you conclude that a line is perpendicular to a plane? Are you allowed to conclude $CD$ is perpendicular $p$ if it is perpendicular to nonparallel lines in $p$? – rschwieb Mar 27 '13 at 21:21

The "plane" and "line" directions determine each other uniquely when perpendicular. If plane $P$ and line $l$ are perpendicular, then any plane $P'$ parallel to $P$ is perpendicular to any line $l'$ parallel to $l$. You are asked to prove the case of one plane $P$ and two parallel lines $l$ and $l'$.
Any proof that two things are perpendicular requires some known set of conditions under which perpendicularity can be inferred (such as a definition of perpendicularity, that you would then hope to show is satisfied). One definition of perpendicularity between a line and a plane is that $\ell \perp P$ if $\ell$ intersects the plane $P$ and is perpendicular to every line in $P$ through the intersection point of $\ell$ and $P$.
There is a canonical line segment joining the intersections of $P$ with the two given parallel lines. A diagram that involves the intersection point with the first line can be translated (pushed without rotation) along that line segment to make a parallel diagram based at the intersection point on the second line. So when checking the criterion for the first line to be perpendicular to $P$, move any perpendicular line involved in the argument at the first intersection point, along the special segment to the second line's intersection point with $P$, and the 90-degree angle formed at the first point will be cloned at the second. Whatever is true at the first location will be transported to the second. That's the logic and, in essence, a complete proof.